presentation of inverse monoids and inverse semigroups

A presentation (for an inverse monoid) is a couple (X;T). We use this couple of objects to define an inverse monoid Inv1⁢⟨X|T⟩. Let ρX be the Wagner congruence on X, we define the inverse monoid Inv1⁢⟨X|T⟩presented by (X;T) as

Inv1⁢⟨X|T⟩=(X∐X-1)∗/(T∪ρX)c.

In the previous dicussion, if we replace everywhere (X∐X-1)∗ with (X∐X-1)+ we obtain a presentation (for an inverse semigroup)(X;T) and an inverse semigroup Inv⁢⟨X|T⟩presented by (X;T).

A trivial but important example is the Free Inverse Monoid [resp. Free Inverse Semigroup] on X, that is usually denoted by FIM⁢(X) [resp. FIS⁢(X)] and is defined by